Dijkstra’s algorithm N: set of nodes for which shortest path already found Initialization: (Start with source node s) n N = {s}, D s = 0, “s is distance zero from itself” n D j =C sj for all j  s, distances of directly-connected neighbors Step A: (Find next closest node i) n Find i  N such that n D i = min Dj for j  N n Add i to N n If N contains all the nodes, stop Step B: (update minimum costs) n For each node j  N n D j = min (D j, D i +C ij ) n Go to Step A Minimum distance from s to j through node i in N

2 Dijkstra's Shortest Path Algorithm Find shortest path from s to t. s 3 t

3 Dijkstra's Shortest Path Algorithm s 3 t        0 distance label S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t }

4 Dijkstra's Shortest Path Algorithm s 3 t        0 distance label S = { } PQ = { s, 2, 3, 4, 5, 6, 7, t } delmin

5 Dijkstra's Shortest Path Algorithm s 3 t    14  0 distance label S = { s } PQ = { 2, 3, 4, 5, 6, 7, t } decrease key  X   X X

6 Dijkstra's Shortest Path Algorithm s 3 t    14  0 distance label S = { s } PQ = { 2, 3, 4, 5, 6, 7, t }  X   X X delmin

7 Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t }  X   X X

8 Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t }  X   X X decrease key X 33

9 Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2 } PQ = { 3, 4, 5, 6, 7, t }  X   X X X 33 delmin

10 Dijkstra's Shortest Path Algorithm s 3 t    14  0 S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t }  X   X X X X X 32

11 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 6 } PQ = { 3, 4, 5, 7, t }  X   X X 44 X delmin  X 33 X 32

12 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t }  X   X X 44 X 35 X 59 X 24  X 33 X 32

13 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 6, 7 } PQ = { 3, 4, 5, t }  X   X X 44 X 35 X 59 X delmin  X 33 X 32

14 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t }  X   X X 44 X 35 X 59 XX 51 X 34  X 33 X 32

15 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 6, 7 } PQ = { 4, 5, t }  X   X X 44 X 35 X 59 XX 51 X 34 delmin  X 33 X 32 24

16 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t }  X   X X 44 X 35 X 59 XX 51 X X 50 X 45  X 33 X 32

17 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 5, 6, 7 } PQ = { 4, t }  X   X X 44 X 35 X 59 XX 51 X X 50 X 45 delmin  X 33 X 32

18 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t }  X   X X 44 X 35 X 59 XX 51 X X 50 X 45  X 33 X 32

19 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 4, 5, 6, 7 } PQ = { t }  X   X X 44 X 35 X 59 XX 51 X 34 X 50 X 45 delmin  X 33 X 32 24

20 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { }  X   X X 44 X 35 X 59 XX 51 X 34 X 50 X 45  X 33 X 32

21 Dijkstra's Shortest Path Algorithm s 3 t   14  0 S = { s, 2, 3, 4, 5, 6, 7, t } PQ = { }  X   X X 44 X 35 X 59 XX 51 X 34 X 50 X 45  X 33 X 32

Modified Dijkstra’s algorithm Dijkstra-aux (G, target-node,sub-path) N: set of nodes for which shortest path already found Initialization: Start with node s= (pop sub-path)//last node on sub-path n V’ = V – {sub-path} //search over nodes not already in sub-path n N = {s}, D s = 0 for s  sub-path, “s is distance zero from itself” n D j =C sj for all j  V’, j  s, distances of directly-connected neighbors Step A: (Find next closest node i) n Find i  N such that n D i = min Dj for j  N n Add i to N n If N contains j=target-node, – return N, C sj – Else return  //no path to target-node Step B: (update minimum costs) n For each node j  N n D j = min (D j, D i +C ij ) n Go to Step A

Modified Dijkstra’s k-Path algorithm Dijkstra-recurse (G, target-node, Path, count) n Do while count< k and Path   – New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to target- node  If New-Path   //another min-cost path –count=count+1; Path-set=Path-set  New-Path –E’ = E – {(pop-Path, target-node)//remove edge from graph –New-Path=Dijkstra-aux (G(V,E’), target-node, pop-Path, count) // graph with edge deleted to prevent finding same path  Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path, count) n End while n Return Path-set

Modified Dijkstra’s k-Path algorithm Dijkstra (G, target-node) Initialization: Start with node s= source node n V’ = V – {s} //search over all nodes n Path-set =  //set of min-cost paths n count=0 //path counter n Path = {s} n Do while count< k and Path   – New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to target-node  If New-Path   //another min-cost path –count=count+1; Path-set=Path-set  New-Path –E’ = E – {(pop-Path, target-node)//remove edge from graph –New-Path=Dijkstra-aux (G(V,E’), target-node, pop-Path )// min-cost path to target-node  Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path ) n End while n Return Path-set

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Modified Dijkstra’s k-Path algorithm Dijkstra-recurse (G, target-node, Path, count) n Do while count< k and Path   – New-Path = Dijkstra-aux (G, target-node, Path)// min-cost path to target- node  If New-Path   //another min-cost path –count=count+1; Path-set=Path-set  New-Path  Else New-Path=Dijkstra-aux (G(V,E), target-node, pop-Path, count) n End while n Return Path-set